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Question

The coefficient of the term independent of x in the expansion of (x+1x2/3x1/3+1x1xx1/2)10

A
70
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B
112
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C
105
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D
210
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Solution

The correct option is D 210
Given expression=(x1/3)3+(1)3(x2/3x1/3+1)(x1)x1/2(x1/21)
=(x1/3+1)+(x2/3x1/3+1)x2/3x1/3+1(x1/2+1)(x1/21)x1/2(x1/21)
=(x1/3+1)(1+x1/2)=x1/3x1/2
(x+1x2/3x1/3+1x1xx1/2)10
=(x+1x2/3x1/3+1x1xx1/2)10
=(x1/3x1/2)10
Tr+1in(x1/3x1/2)10is
10Cr(x1/3)10r.(1)r.(x1/2)r
=(1)r10Crx(10r3r2)which is independent of x
If (10r3r2)=0r=4
Hence required coefficient =10C4(1)4=210

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